Rohit Pandey
This paper presents a new proof of the renewal theorem by bijecting a general point process to a deterministic one (where the time between events is always fixed). It also provides insight into the workings of the renewal theorem.
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Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and its extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell's renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give condit…
Generalized Fibonacci Numbers and Blackwell's Renewal Theorem
研究了广义斐波那契数与随机过程更新理论之间的联系。利用Blackwell更新定理，我们找到了广义斐波那契数的一个近似.借助于更新定理中的误差估计，我们得到了一个显式表示。
From uniform renewal theorem to uniform large and moderate deviations for renewal-reward processes
从一致Blackwell更新定理出发，导出了一致密钥更新定理.在比指数矩存在性弱得多的条件下，得到了更新-报酬过程的一个一致的大偏差原理（LDP），并导出了中等偏差原理（MDP）.
An elementary renewal theorem and a Blackwell theorem provided by Jasiulis-Gołdyn et al. (2020) in a setting of Kendall convolutions are proved under weaker hypothesis and extended to the Gamma class. Convergence rates of the limits concerned in these theorems are analyzed.
The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem for renewal processes defined by Kendall random walks.
Blackwell-type Theorems for Weighted Renewal Functions
对于满足其“平均行为”的广泛假设的数字序列${a_n}$和具有独立同分布的随机游走$S_n=ξ_1 +...+ξ_n$本文讨论了具有正均值$μ$的$ξ_j$跳跃，建立了[ \sum_{n\ge 1} a_n \pr（S_n \in [x，x+ \D））\quad为\quad × \to \infty，]和的渐近行为，其中$\D>0$是固定的.我们的结果的新颖之处不仅在于在更广泛的条件下的权重${a_n}$，但也在既不是跳跃$ξ_j$和权重$a_j$需要是积极的。证明的关键工具是积分局部极限定理和大偏差界.对于跳分布$F$，我们考虑了四种条件：（a）$F$的二阶矩是有限的，（b）$F$属于稳定律的吸引域，（c）$F$的尾属于所谓的局部正则变化函数类，（d）$F$满足矩Cramér条件.关于权重，在情况（a）-（c）中，我们假设${a_n}$是所谓的$ψ$ -局部平均常数序列，$ψ(n)$是确保$(S_n - μn)/ψ(n)$的分布收敛到相应稳定律的比例因子。在情况（d）中，我们考虑$a_n=b_n e^{qn},$形式的权重序列，其中${b_n}$具有关于$ψ(n)=\sqrt{n}.$的情况（a）-（c）中的序列${a_n}$假设的性质
On renewal theory for cluster processes
本文研究带标记与簇的更新过程，利用耦合方法证明 Blackwell、关键、扩展及初等更新定理的推广形式在标记与等待时间存在依赖时仍成立。
Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\cdots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. Further, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For $j\in\mathbb{N}$ and $t\…
Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T:=(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\ldots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. Consider a general branching process generated by $T$ and denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. We treat early generations, that is, fixed generations $j$ which do not depend on $t$. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell …
Based on a simple object, an i.i.d. sequence of positive integer-valued random variables, $\{a_n\}_{n\in \mathbb{Z}}$, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time $n$ and dies at time $n+a_n$. This dynamics is that of a D/GI/$\infty$ queue, with arrivals at integer times and service times given by $\{a_n\}_{n\in \mathbb{Z}}$. Second, the directed random graph $T^f$ on $\mathbb{Z}$ generated by the random map $f(n)=n+a_n$. Only assuming $\mathbb{E}[a_0]<\infty$ and $\mathbb{P}[a_0=1]>0$, we show that,…
Multi-time Markov renewal chains and stratified renewal theorems
研究在标准博雷尔状态空间上的离散马尔可夫更新理论，利用向量值逗留时间等。通过马尔可夫更新势提供多种表示，确定半马尔可夫场的马尔可夫增强及准则，建立分层逆更新理论并给出相关极限与估计，连接了多方面内容。
Limit theorems for multidimensional renewal sets
考虑由i.i.d.生成的$d$维整数网格上的多个和$S_n$。\正期望值的随机变量。证明了随机集${\mathcal M}_t$作为多重和的逆的强大数律、重对数定律和分布极限定理，即作为所有自变量$x\in{\mathbb R}_+^d$的集合，使得内插的多重和$S_x$超过$t$ .该矩条件与多重和的几乎处处极限定理中的条件相同。结果表示在集合包含和使用集合之间的距离。
A Proof of Basic Limit Theorem of Renewal Theory
本文针对更新理论中 Blackwell 更新定理的离散情形，基于 Fourier 分析方法与几乎收敛序列的 Tauber 型定理，给出了极限 $\lim_n p_n=1/\sum_{i=0}^\infty iq_i$ 的一个新证明，并指出实际仅需初等分析。
Binomial ARMA count series from renewal processes
本文提出了一种由更新过程生成平稳整值时间序列的新方法。证明了如果更新过程的寿命分布是非格的，且概率母函数是有理的，则生成的时间序列满足因果可逆的ARMA型随机差分方程.这一结果为生成具有ARMA型自协方差函数的整值时间序列提供了一种简便的方法。作为例子，给出了从滞后p后具有常数风险率的寿命分布生成二项式ARMA（p，p-1）序列的例子。提出了AR（p）情形的估计方法。
We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the ex…
How can a clairvoyant particle escape the exclusion process?
我们研究的检测问题，在以下设置：在一维整数格，在时间为零，放置在每个站点上的节点独立的概率$ρ\in [0,1)$，让他们演变为一个简单的对称排它过程。在时间零点，在原点放置一个目标。目标仅以整数倍移动，并且可以移动到距其当前位置$R$距离内的任何站点。还假设目标可以预测所有节点的未来移动。我们证明，对于足够大的$R$（取决于值的$ρ$），它是可能的目标，以避免检测永远与积极的概率。这个结果的证明使用了两个独立的成分。首先，我们建立了一个重正化方案，可以用来证明渗流依赖定向模型在一定的解耦条件。这个结果是一般性的，不依赖于模型的特性。作为应用，我们证明了我们的主要定理不同的动力学，如独立的随机游动和独立的更新链。我们还证明了存在的定向渗流的随机交错和其空集的大尺寸。证明的第二步是排它过程的时空解耦。
This article presents a limit theorem for the gaps $\widehat{G}_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of a sample of size $n$ from a random discrete distribution on the positive integers $(P_1, P_2, \ldots)$ governed by a residual allocation model (also called a Bernoulli sieve) $P_j:= H_j \prod_{i=1}^{j-1}(1-H_i)$ for a sequence of independent random hazard variables $H_i$ which are identically distributed according to some distribution of $H \in (0,1)$ such that $- \log(1 - H)$ has a non-lattice distribution with finite mean $μ_{\mbox{log}…
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